5.9 Inhomogeneous models

Homogeneous models provide a rich generalization of isotropic ones, but inhomogeneities lead to
stronger qualitative differences. To start with, at least at the kinematical level one has infinitely many
degrees of freedom and is thus always dealing with field theories. Studying field theoretical implications does
not require going immediately to the full theory since there are many inhomogeneous models of physical
interest.

We will describe some 1-dimensional models with one inhomogeneous coordinate and two others
parameterizing symmetry orbits. A general connection is then of the form (with coordinate differentials
and depending on the symmetry)

with three real functions and three internal directions normalized to , which
in general are independent of each other. The situation in a given point is thus similar to general
homogeneous models with nine free parameters. Correspondingly, there are not many simplifications from
this general form, and one needs analogs of the diagonalization employed for homogeneous models.
What is required mathematically for simplifications to occur is a connection with internally
perpendicular components, i.e., in each point. This arises in different physical
situations.